Convolution of hp-functions on locally refined grids

Abstract

Usually, the fast evaluation of a convolution integral ∫ ℝ f(y)g(x - y)dy requires that the functions f and g have a simple structure based on an equidistant grid in order to apply the fast Fourier transform. Here, we discuss the efficient performance of the convolution of hp-functions in certain locally refined grids. More precisely, the convolution result is projected into some given hp-space (Galerkin approximation). The overall cost is O(p 2 N log N), where N is the sum of the dimensions of the subspaces containing f , g and the resulting function, while p is the maximal polynomial degree.